An Algorithm for Persistent Homology Computation Using Homomorphic Encryption
Bibliographic record
Abstract
Topological Data Analysis (TDA) provides a suite of tools that extract shape-based features from high-dimensional data, with applications to modern statistical and machine learning (ML) models. Among these tools, persistent homology (PH) summarizes the topological structure of data in compact representations known as persistence diagrams (PDs). Due to their robustness to noise, interpretability, and compatibility with standard ML architectures, PDs are increasingly used in applications involving sensitive data, such as genomics, cancer research, sensor networks, and finance. Thus, there is a growing need to incorporate TDA methods into secure, end-to-end data analysis pipelines. We present the first adaptation of a fundamental TDA algorithm known as boundary matrix reduction to operate on encrypted data using homomorphic encryption (HE). We provide mathematical guarantees for the correctness of the HE-compatible algorithm under appropriate parameter choices and analyze its computational complexity. We support these theoretical results with two distinct empirical studies: (1) a plaintext simulation that explores the extent to which the theoretically sufficient parameters can be relaxed while still preserving correctness, and (2) a working implementation in the OpenFHE framework that validates correctness on encrypted data. This work lays the foundation for fully encrypted topological computations and opens new directions in privacy-preserving data analysis using TDA.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".